Susan, driving north at 60 mph, and Trent, driving east at 45 mph, are approaching an intersection. What is Trent's speed relative to Susan's reference frame?

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Define the problem in terms of relative velocity. The relative velocity of Trent with respect to Susan is the vector difference between Trent's velocity and Susan's velocity: \( \vec{v}_{TS} = \vec{v}_T - \vec{v}_S \).

Express the velocities of Susan and Trent in vector form. Since Susan is driving north at 60 mph, her velocity vector is \( \vec{v}_S = 60 \hat{j} \) (where \( \hat{j} \) represents the northward direction). Trent is driving east at 45 mph, so his velocity vector is \( \vec{v}_T = 45 \hat{i} \) (where \( \hat{i} \) represents the eastward direction).

Subtract Susan's velocity vector from Trent's velocity vector to find the relative velocity: \( \vec{v}_{TS} = \vec{v}_T - \vec{v}_S = 45 \hat{i} - 60 \hat{j} \).

Calculate the magnitude of the relative velocity vector using the Pythagorean theorem: \( |\vec{v}_{TS}| = \sqrt{(45)^2 + (-60)^2} \).

Interpret the result. The magnitude \( |\vec{v}_{TS}| \) represents Trent's speed relative to Susan's reference frame, and the direction can be determined using trigonometry if needed.

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주요 개념

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Relative Velocity

Relative velocity is the velocity of one object as observed from another object's frame of reference. It is calculated by subtracting the velocity vector of the observer from the velocity vector of the object being observed. In this case, to find Trent's speed relative to Susan, we need to consider both their velocities and the direction they are traveling.

Vector Addition

Vector addition is a mathematical operation used to combine two or more vectors to determine a resultant vector. In the context of relative velocity, this involves adding the velocity vectors of Susan and Trent, taking into account their directions. Since they are traveling at right angles to each other, the Pythagorean theorem is often used to find the magnitude of the resultant vector.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this scenario, it helps calculate the resultant speed of Trent as observed from Susan's perspective, as their paths form a right triangle.